The Demand function (L3)
Discussion question
the price of beans went up, so people started buying more beans
Could this ever be true? Is there a logical economic justification for this?
Discuss with your neighbour… we will vote
Key goals
Key goals of these lectures (and accompanying self-study)
How do you derive an individual’s demand curve from her utility function?
- What causes shifts in either?
What properties do demand curves have?
Understand the following concepts/outcomes (& how to derive them):
Substitution and income effects (of a price change)
Goods that are ‘substitutes’ or ‘complements’ (WARNING!)
Consumer surplus (from a transaction)
The lump-sum principle (and the distortion of taxation)
- What is a market demand curve, and how do we derive it?
Understand the concepts:
Price elasticity (of market demand for a product), and what it means to firms’ pricing strategy
Income elasticity (…)
Cross-price elasticity (between two products)
- Be able to discuss real world examples and applications of the above
Understand real-world issues:
A ‘Fixed-basket’ consumer price index (CPI) may overstate inflation
The lump-sum principle, the distortion of taxation
Fundamental concepts, useful for business & policy:
Goods may be ‘substitutes’ or ‘complements’ for one another (not the same as ‘perfect’ substitutes/complements!)
How can we consider/compute the Consumer surplus from a transaction?
Demand functions
Demand functions
- Previous: consumption choices determined by utility functions/indifference curves and budget constraints
\[Quantity \: of \: X \: demanded = d_x(P_X, P_Y, I; preferences)\]
Homogeneity
- Homogeneous (of degree zero) (demand) function
- A function whose outcome value does not change when all arguments are changed proportionally
\(d_X(P_X,P_Y,I)\) is homogenous in its arguments
- Multiply all prices and income by same amount \(\rightarrow\) budget constraint unchanged \(\rightarrow\) consumption choices the same
E.g., \(P_X X + P_Y Y = I\) is the same as \(2P_X X + 2P_Y Y = 2I\)
Response to income changes
\[d_x(P_X, P_Y, \mathbf{I}; preferences)\]
- What happens to the quantity purchased of some good as your income increases?
- Depends on whether the good is normal or inferior
\[d_x(P_X, P_Y, \mathbf{I}; preferences)\]
- Normal good
- A good that is bought in greater quantities as income increases.
- Inferior good
- A good that is bought in smaller quantities as income increases.
Inferior good
Inferior good
Move to PPT here … “Changes in Income: A Normal Good”
\(\Delta\) price \(\rightarrow\) substitution & income effects
\[d_x(\mathbf{P_X}, P_Y, I; preferences)\]
\[d_x(\mathbf{P_X}, P_Y, I; preferences)\]
- Substitution effect (‘Hicksian’)
- The effect on consumption due to a change in price ‘holding utility constant.’
Precisely: effect on the lowest-cost bundle yielding this utility
- Income effect (of a price change)
- The remaining effect on consumption; price change \(\rightarrow\) change in purchasing power/achievable utility.
income and substitution effects
(ppt here… “Change in a good’s price”)
- Substitution effect: ALWAYS opposite direction as price change
- Income effect
- Normal good \(\rightarrow\) Opposite direction as price change
- Inferior good: Same direction as price change . . .
Thus
- Normal good: Substitution & income effects go in same direction
- Inferior good: Substitution & income effects go in opposite directions
- \(\rightarrow\) Net effect unknown
- Usually substitution effect dominates but see Giffen goods
Read on your own, know:
- Numerical example of response to price change
- The relative importance of substitution effects for most goods
- Substitution and income effects for inferior goods
Different substitution effects
Perfect complements: No substitution effect, only an income effect of a price change
Perfect substitutes: Large substitution effect – price change may cause a complete switch
In between: depends on curvature of indifference curve
The (legendary?) Giffen good
- If the price of a good increases, can quantity demanded actually increase?!
- Yes, if the good is very inferior and is a large portion of income
Motivating example… WHAT???!!
Hands up please (or turningpoint if we’ve time: A=no, B-yes). If you could not resell or gift it…
- Would you accept a free IPhone 11 pro?
Would you pay:
- 250?
- 500?
- 750?
- 1000?
- 1250?
If your income was 100,000, would you pay
Would you pay:
- 250?
- 500?
- 750?
- 1000?
- 1250?
Back to your current income .. what if all other smartphones cost 2000?
Would you pay for the Iphone 11pro:
Would you pay:
- 250?
- 500?
- 750?
- 1000?
- 1250?
What do these things have in common?
What explains this?
\[Quantity \: of \: x \: demanded = d_x(p_x, p_y, I; preferences)\]
Previous lecture: What the constrained util-max implied for…
‘Homogeneity of degree zero’ of \(d_x(p_x, p_y, I)\)
How \(d_x\) responds to \(I\) (remember: inferior & normal goods)
How \(d_x\) responds to \(P_x\) (rem: normal or Giffen)
- Substitution and income effects
Goals: This lecture chunk (demand part 2)
Util-max s.t. constraints \(\rightarrow\)
Understand real-world issues:
‘Fixed-basket’ CPI may overstate inflation
Lump-sum principle, distortion of taxation
Fundamental concepts, useful for business & policy:
Goods that are ‘substitutes’ or ‘complements’
Consumer surplus (from a transaction)
…
Derive
Individual’s demand curve from her utility function
Market demand curve
- What causes shifts in either?
- How to compare (the price and income elasticities) of apples and orange juice
App 3.2: The CPI and it’s biases
App 3.2: The CPI and it’s biases
A very important number: Used for monetary policy and for targeting many salaries and benefits
- But does it overstate the rate of inflation?
Based on a ‘typical market basket’
- UK: of 700 different goods and services, excluding housing, updated yearly.
A good example, 1982 vs 2012:
\[b_{82}=p^x_{82}x_{82}+p^y_{82}y_{82}\] \[b_{12}=p^x_{12}x_{82}+p^y_{12}y_{82}\] \[cpi_{12}=\frac{b_{12}}{b_{82}}\]
The Lump-Sum Principle
Have you seen this?
What is going on here?
- The social cost (deadweight loss) is greater the more taxes change ‘compensated’ behaviour (via substitution)
- The most efficient tax:
- raises the most revenue for a given utility loss
- reduces utility the least for a given revenue
- … is a ‘lump-sum tax’: same tax no matter what you do (including work/leisure!)
- rationale for the poll tax
FT Comment: New rule may even reduce the tax take
Suppose I am willing to sell a widget I manufacture for £10 and that you are willing to spend £11 to own it. Then we will do business and I will receive a consumer surplus of £1. Now suppose the government imposes a 20 per cent sales tax, and the price I charge for widgets increases to £12. Since £11 is your maximum price, you won’t buy it and the £1 of consumer surplus you would have gained is lost.
This is how tax avoidance imposes a so-called deadweight cost on society. People avoid doing valuable things that they would have done if not for the tax.
Read on your own, know:
Potential inefficiency of in-kind programmes and subsidies (App 3.3)
- On the other hand, the benefits of in-kind programmes rather than cash transfers
Changes in the Price of Another Good
Changes in the Price of Another Good; substitutes and complements
- Complements
- If rise in \(p_x\) \(\rightarrow\) \(q_{d,y}\) decreases (& v/v), goods y and x are (gross) complements to one another.
These ‘cross-price effects’ include both *substitution and
- Substitutes
- If \(p_x\) \(\uparrow\) \(\rightarrow\) \(q_{d,y}\) \(\uparrow\) (and vice-versa), goods y and x are (gross) substitutes …
Warning: \(\neq\) ‘perfect’ complements/substitutes
Individual demand curves
Individual demand curves
\[d_x(p_x,p_y,I; \: preferences)\]
- ‘Individual demand curve’: plots individual’s purchase of a good versus it’s price
‘Map it out’: increase \(p_x\) \(\rightarrow\)
budget constraint shifts inwards \(\rightarrow\)
- New point tangent to indifference curve
Shifts in an individual’s demand curve
What might cause an individual’s demand curve for a product to shift (inward or outward)?
- Not ‘a change in the price of that good’.
- Not a shift in the supply curve.
Yes:
- Change in price of complements or substitutes for that good
- Change in income
- Maybe: \(\Delta\) consumer’s info, preferences, weather, etc.
Consumer surplus
- Consumer surplus
- The extra value individuals get from consuming a good over what they pay for it.
My wtp for ‘right to consume good at its current price’ (rather than not at all)
- Measures consumer welfare, used for policy analysis
- Area between demand curve and market price:
Market demand
Market demand
- Market demand
- Total quantity of a good demanded by all consumers
Sum individual quantities demanded (at a given price)
- Market demand curve
- Relationship between total quantity demanded of a good and its price, ceteris paribus
Market demand curve: Relationship between total quantity demanded of a good and its price, ceteris paribus.
Sum the individual demand curves ‘horizontally’ … quantities demanded at each price
Shifts in the Market Demand Curve
Similar things that cause individual demand curve shifts:
- Increases in overall income (for normal goods)
- Reduced prices of complements, increased price of substitutes
A random example:
Elasticities
Elasticities: comparisons across contexts
Which is ‘larger’?:
change in \(q_d\) of oranges when \(p_{orange}\) rises or
the change in \(q_d\) of apple juice when its price rises?
Difficulty: Measured in different units, p and q have different starting values!
- Elasticity
the measure of the % change in one variable brought about by a 1% change in another variable.
- a unitless measure; will be the same no matter how these variables are measured.
If a 5% fall in the price of oranges typically results in a 10% increase in quantity bought
we might say that ‘each percent fall in the price of oranges leads to an increase in sales of about 2 percent’
i.e., the ‘elasticity’ of orange sales wrt price is about 2 . . .
But elasticities need not be constant; they may depend on starting point
E.g., linear demand \(\rightarrow\) different price elasticity at each point
Price elasticity of demand
Price elasticity of demand: \[e_{Q_d,p} = \frac{percent \ change \ in \ Q_d}{percent \ change \ in \ p} \] \[ = \frac{\Delta Q_d}{Q_d}/\frac{\Delta p}{p}\]
(actually, the limit of this as these changes converge to zero)
Should always be negative (except for Giffen goods)
A unitless measure related to the slope of the demand curve
Very important for price-setting firms (more on this later)
Examples from the headlines
India’s Hike Messenger takes aim at WhatsApp
‘Reliance ended up showing that there is elasticity in the market. If you drop prices, people will come on board,’ he said.
Next to add more space despite retail sales ‘moving backwards’
The retailer does not expect any impact from the drop in sterling since the Brexit vote to kick in until at least the spring of 2017, as it had hedged some of its foreign-currency exposures in advance. Still, it expects expenses to rise by up to 5 per cent next year. `The last time we had to increase prices (which was in 2010 when cotton prices soared) we estimated that price elasticity was around 1.1.’
If that remains the case today, a retail selling price increase of 5% would result in a fall in unit sales of -5.5% and a fall in like for like sales value of between -0.5% to -1.0%. In the scheme of things, we think that this drag on sales is manageable and less damaging than taking a significant hit to margin.’
Properties of price elasticity of demand:
- Goods w/ close substitutes at a close price \(\rightarrow\) highly elastic
- … with few substitutes … inelastic
- Typically: elasticity greater in long run than short run. Why?
| \(e_{Q,p}\) | \(abs(e_{Q,p})\) | Term |
|---|---|---|
| \(< -1\) | \(>1\) | Elastic |
| \(= -1\) | \(=1\) | Unit Elastic |
| \(> -1\) | \(<1\) | Inelastic |
- Total expenditure (revenue): price \(\times\) quantity
- So percent change in total expenditure is:
- pct change price \(\times\) pct change quantity
As \(e_{Q,p}\) tells you the pct change quantity for a small pct change in price:
| \(abs(e_{Q,p})\) | Term | p rise \(\rightarrow\) expdr | p falls \(\rightarrow\) expdr |
|---|---|---|---|
| \(>1\) | Elastic | Falls bc Q falls more | Rises |
| \(=1\) | Unit Elastic | Constant | Constant |
| \(<1\) | Inelastic | Rises bc Q falls less | Falls |
More to read
- Numerical example (may be covered in tutorial)
- Skip: Unit Elastic Curve
- Read on your own: Application 3.7: An Experiment in Health Insurance
Income elasticity of demand
Income elasticity of demand
- Income elasticity of demand
- % change in quantity demanded of a good in response to 1% change in income.
\[e_{Q_d,I} = \frac{percent \ change \ in \ Q_d}{percent \ change \ in \ I} \] \[ = \frac{\Delta Q_d}{Q_d}/\frac{\Delta I}{I}\]
Normal goods: \(e_{Q,I} > 0\)
Inferior goods: \(e_{Q,I} < 0\)
Luxury goods: \(e_{Q,I} > 1\)
Prof. Muellbauer letter to FTo
Sir, Professor Gordon Gemmill (Letters, December 14), surprisingly for a trained economist, assumes an income elasticity of demand of zero for housing: that is, that people do not demand more and better housing as they become richer. Nowhere in the world is this the case! My own empirical work demonstrates that around two-thirds of the rise in UK house prices, corrected for general inflation, since 1980 is because supply is not keeping up with income and population growth.
Other drivers do exist … The price effects of extra supply take time to build up. I agree on that. But just imagine what would happen if we did nothing more than we are now doing: population and income growth would drive prices even higher even though we already hold the record for rises in house prices since 1970 among the group of seven leading high-income countries. We need to build far more housing, in the right locations. And we need to start now. - Prof John Muellbauer Nuffield College, Oxford, UK
Rest of NS chapter 3
Read on your own:
Cross-price elasticity of demand (read on your own)
Some elasticity estimates (note these are a bit dated)
Other practice questions
Other practice questions
Does this make sense??
Our company is very good at motivating our employees. We get twice as much production for each pound we spend on salaries as for each pound we spend on better machines; and we continue to invest in both.
???
We need to keep producing this product, even though it’s unpopular, because it cost us £1 million to develop, and we need to make this back.
???
My business is making a profit, even though I could have earned more money putting the £1,000,000 in a savings bond.
???